computer algebra systems (cas): from where did they come, and where might they go? ed dickey...
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Computer Algebra Systems (CAS): From Where Did They
Come, and Where Might They Go?
Ed Dickey
University of South Carolina
National Council of Teachers of Mathematics
Annual Meeting – San Diego
22 April 2010
Jacques Barzun, Teacher in America
“ I have more than an impression- it amounts to a certainty- that algebra is made repellent by the unwillingness or inability of teachers to explain why…
There is no sense of history behind the teaching, so the feeling is given that the whole system dropped down ready-made from the skies, to be used by born jugglers.”
Plan for this Session
• A Look Back…– General and Personal History of CAS– Contrast with dynamic geometry– Input from key researchers and educators
• A Look Forward…– What needs to happen for CAS to have an impact on
school mathematics– Views from key mathematicians, mathematics
educators and researchers, developers, and teachers
My First CAS Experiences…
• NSF Conference Report, October, 1982
• Graphing utilities, VisiCalc, TK!Solver, and muMATH in school algebra
From 1982…
• “A new role for manipulative skills.” As calculators impact arithmetic, “computers seem to offer the same promise in algebra; diminishing the importance of developing student skill in algebraic manipulations.”
• “The preceding proposals of topics to be deleted or given reduced attention in high school algebra will certainly provoke vigorous dissent from all corners of the mathematical community.”
My First CAS Experiences…
“The biggest need at the moment, in my opinion, is to have a good, thorough look at the total elementary and secondary curriculum… in mathematics, to see how the priorities of topics, and pedagogical possibilities, and the interaction among the topics change in light of current technological possibilities.. How do they change what we ought to do?”
Henry Pollak (interviewed in Steen and Albers, 1981 and quoted p.3 of Computing and Mathematics)
Congratulations!
Lifetime Achievement Award for Distinguished Service to
Mathematics Education
To Professor Pollak for the 2010 Mathematics Education Trust
My First CAS Experiences…
• 1984 from NCTM Year Book
• “Imperatives and Possibilities for New Curricula in Secondary School Mathematics” by Fey and Heid
• Page 23
My First CAS Experiences…
• 1986 NCTM Charleston Regional
• Why and How to Use Symbol Manipulation Software
• muMATH-80 for the Apple II
My First CAS Experiences…
• Phone call to Albert Rich and Dave Stoutemyer
• 1987 USC became “world distributor” of muMATH-80
• Teacher’s Guide
• $40 ($5 Soft Warehouse royalty)
Smithsonian Exhibit
• Slates, Slide Rules, and Software: Teaching Math in America
• muMATH, Derive, TI-92
History of CAS
• LISP in 1962
• Physicist Martin Veltman’s Schoonschip (“clean ship” in Dutch) in 1963
• U.S. Carl Engelman’s MATHLAB in 1964 led to Macsyma
CAS in Schools
“Computer Symbolic Math & Education: A Radical Proposal” ACM SIGSAM Bulletin 13, 3 (August 1979), pp. 8-24.
CAS in Schools
NCTM 1989 Standards
The Information Society. … The use of this technology has dramatically changed the nature of the physical, life, and social sciences; business; industry; and government. ….
The new technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them.
NCTM 1989 Standards
• The use of computer utilities to develop conceptual understanding
• The use of technology in instruction should further alter both the teaching and the learning of mathematics.
• Computer software can be used effectively for class demonstrations and independently by students to explore additional examples, perform independent investigations, generate and summarize data as part of a project, or complete assignments.
South Carolina Standards
Elementary Algebra A-1.7 [2007]
• Understand how to represent algebraic relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs)
• Also mentioned in Intermediate Algebra and Precalculus.
CAS in Schools: Science Fair
Even among mathematicians…
• Even this year, a former editor of a SIAM journal and current math department chair did not realize a CAS could compute 1000! in full precision.
• Have you seen it? All 2,658 digits?
• Thank you, Derive 5!
Mathematics Teacher, Sept 1989
“How Symbolic Mathematical Systems Could and Should Affect Precollege Mathematics.” M. Kathleen Heid
• muMATH, Derive, IBM Math Exploration Toolkit, HP28S
• “… school mathematics may be able to progress toward better understanding of concepts, mathematical modeling, superprocedures, numerical and symbolic patterns and the equivalence of mathematical representations.”
Textbook 1990s
• Fey, Heid, Good, Sheets, Blume, Zbiek, Janson Publications,1995
Table of Contents1. Variables and Functions 2. Functions and Computing Technology 3. Linear Functions 4. Quadratic Functions 5. Exponential Functions 6. Rational Functions 7. Systems of Functions and Equations 8. Symbolic Reasoning : Equivalent Expressions
9. Symbolic Reasoning : Equations and Inequalities
Textbook 2009
Al Cuoco and others with EDC• “platform for experimenting” • “reducing computational overhead”• “build computational models”
Dynamic Geometry
• Geometric Supposer, 1985,Judah Schwartz and Michal Yerushalmy
Dynamic Geometry
• Cabri, 1994, Jean-Marie Laborde
• Geometer’s Sketchpad, 1995, Nick Jackiw
• “Lifting the Curtain: The Evolution of The Geometer’s Sketchpad” by Daniel Scher
Graphing Utilities
• Widespread use
• 1985 Casio 7000G
• Handheld CAS
Ubiquitous access…
• Wolfram Alpha, free on web
• iPhone Symbolic Calculator $.99
• PocketCAS $4.99
Some 40 years have passed…
• 25 years since my own first exposure to CAS
• … and SO WHAT?
• What can teachers, mathematicians, mathematics educators, researchers, and developers tell us about…
• Where we are today…
• And where we might go tomorrow
Method
• Wrote to 35 of the “best and brightest” involved in CAS work for at least the past 15 years.
• Four Questions about CAS
• 19 generous responses
Mathematicians
• Lynn Steen
• Tom Dick
• Jeanette Palmiter
• Bert Waits
Teachers
• John Mahoney
• Lin McMullin
• Natalie Jakucyn
Researchers
• Paul Goldenberg
• Carolyn Kieran
• Paul Drijvers
• Luis Saldanha
Mathematics Educators
• Jim Fey
• Karen Hollebrands
• Thomas Edwards
• Bob Ronau
• Johnny Lott
Educators/Developers
• Al Cuoco
• Bernhard Kutzler
• Albert Rich
The Questions
1. To what degree has the promise you envisioned when you first became aware of symbol manipulation or computer algebra software been realized in terms of:
a) the development and maturation of the technology;b) the impact on school or college mathematics
curriculum;c) teachers’ decisions to make use of the software for
teaching mathematics; ord) students’ learning of mathematics in relation to the
use of the software?
The Questions
2. How has your concept of symbol manipulation or computer algebra software changed since you first learned about it?
The Questions
3. What development(s) do you foresee related to CAS software technically or as a tool for teaching and learning?
The Questions
4. If you feel the full potential of computer algebra software has NOT been realized, what impediments must be removed or what issues must be addressed for CAS to have a more significant impact on the teaching and learning of school mathematics?
“Promise Realized?”
Steen responded by stating that his initial vision was “a threat to the status quo in mathematics education from roughly grades 9 through 14.
This threat held the potential for stimulating positive change, but also for inhibiting deep understanding….
It’s impact on school and college curricula continues to be a patchwork, which does not surprise me.”
“Promise Realized?”
Palmiter sees “very little change from the early 1990s when many colleges were first embracing CAS… If anything, the move has been to withdraw from using CAS …Teaching with CAS is/was viewed (with good reason) as labor intensive, not lending itself well to large lecture halls or even adjunct/TA taught sections.
I see schools/colleges creating different tracks, one in which CAS is not part of the course (intended for weaker students where they can just spend their time on algebraic manipulations) and another track where CAS is an integral part of the course (intended for stronger students, in which we don’t mind putting a bit more effort).”
Development & Maturation of CAS
• 12 of the 13 responding cited increased capabilities (“power”) and the improved “ease of use.”
• Lott and others cited the “leap forward” coming with the TI-92.
• McMullin saw the mid 90s as a “peak” in CAS use at the high school level.
• Ronau described the maturation as moving from “cumbersome and limited” to “cumbersome and powerful” to “accessible and powerful” where CAS is now available on many platforms with a more natural user interface.
Development & Maturation of CAS
• Waits described the technology as maturing “nicely including hand-held CAS. However, attention to the user-interface lags. More attention needs to be paid to the KISS principle.”
• Kutzler offered a different view: that the development and maturation was not nearly “as much as would have been possible” and offered a “plea” for Pedagogical CAS (PeCAS) in which emphasis is placed on supporting “teachers and students in teaching, learning, and doing mathematics” and replaces CAS with PeCAS by implementing system features that support the pedagogy and facilitate access and application of mathematical (algebraic, numeric, graphic) operations..
Impact on School or College Math
• To the question on impact on school and college curriculum, 11 responded with terms like
• Edwards “fails to meet my expectation.”• Lott “far less” impact • Hollebrands not much “uptake”• Mahoney “appears CAS related change won’t occur”• McMullin “far too little”• Fey “modest” impact…• Goldenberg “In my own work, I don’t see CAS anywhere… Few
curricula seem to know what to do with it (… obvious exceptions like Fey’s work and Cuoco’s).”
• Drijvers “marginal”• Waits “ZIP, NONE, NADA... in the US for school mathematics…
some bright spot... in some regions in Asia, Europe, and Canada.”
Impact on School or College Math
Mahoney summaries reason for lack of impact:
1. “Most teachers are uncomfortable using CAS
2. Most teachers believe that students need to do problems the old way – without CAS.
3. Problems on national tests (PSAT, SAT, AP Calc) have become CAS-proof. In other words, having a CAS unit won’t help a student do the problem. So in some ways, it isn’t particularly useful for a student to have a CAS handheld.”
Impact on School or College Math
• Others (Cuoco, Jakucyn, Lott, McMullin and Drijvers) agree that traditional expectations and assessment issues have worked against impact of CAS in the mathematics curriculum.
• Lott offered that “schools are not using it to the degree that they could be; colleges are not accepting of CAS; and teacher preparation programs have not taught prospective teachers how to use it effectively.”
Impact on Math Curriculum
• Anonymous: “… the policy-powerful traditionalists in mathematics are forcing us to miss (or at least delay) a once in a lifetime opportunity for a great leap forward in empowering all students to learn and come to use really powerful mathematical ideas.
• They won’t even consider experiments to see the potential benefits and possible pitfalls.
• They continue to flog a curriculum perhaps appropriate for some in the 1950’s as the common core expectations for all students.”
Teachers’ Decisions to Use CAS
• 11 responded citing reasons like fear of or lack of familiarity as reasons for not using CAS.
• Also mentioned by some were teachers’ beliefs about mathematics learning.
Teachers’ Decisions to Use CAS
Edwards: most teachers still fear the use of technology to support teaching mathematics. Among those who claim to be open to such use, we are still hearing things like, “… but they have to learn it by hand first.”
Lott: “Many teachers at the high school level are as afraid of CAS as elementary teachers have been of calculator usage in grade schools. Many still will not consider it as a valuable learning option for students. Or maybe worse they will not allow ‘slower’ students to use the CAS, reserving it for use only by the more advanced students. This latter decision is likely the ‘most wrong’ of all.”
Teachers’ Decisions to Use CAS
McMullin: “The state tests drive the curriculum (actually they are the curriculum) and, by and large, teachers will not make use of CAS until and unless the state testing forces them to. Don’t hold your breath.”
Ronau: “If the software improves to the point of being seamless … as mathematical ideas move from representation to representation, and that the process is natural and intuitive, then teachers are likely to embrace this technology, provided that they have assurances that it will not cause their students to falter on whatever assessments are driving their lives.”
CAS Relation to Student Learning
• 12 responded often citing the metaphor of the “black box” vs. “glass box” CAS (pops out an answer with not insight vs. provides insights into the process).
• Also cited was a lack of definitive research on student learning with CAS.
CAS Relation to Student Learning
Lott: “Ironically, I don’t think that we have gotten to the point yet where we know enough about the use of CAS and its affect on student learning.”
Fey: “We really have not studied this carefully enough.”
Ronau mentions the current “learning curve for CAS being a true multi-representational tool is too steep. “Students are not driving yet. Again, the software is not seamless or intuitive yet.”
CAS Relation to Student Learning
Dick discusses the black-box metaphor: “… the main advantage offered by CAS as seen by many proponents and opponents is CAS as servant to whom we can delegate computational tasks
(proponents might think of tasks that are tedious or out of reach for paper-and-pencil; opponents might think of tasks that they feel SHOULD be done by paper-and-pencil).”
Waits adds “the answer box mentality is seen all too frequently and inappropriate use is a real problem…”
CAS Relation to Student Learning
Drijvers mentions that “the math education research community and the math educators community have not succeeded in convincing authorities and ‘ordinary’ teachers of the benefits of integrating computer algebra.
So far, we have not really tackled the issue of the complex relationship between by-hand skills, insight and tool use, and this makes it easy for the rest of the world to neglect computer algebra.”
• Kieran and Drijvers offer insights to deal with these issues based on their research.
CAS Relation to Student Learning
Kieran: “I have observed many teachers… have a tendency to focus on the technology and not on the mathematics. Those who succeed with enriching students’ learning of mathematics, in conjunction with this [CAS] technology, are generally those who have a vision of the way in which the technology can be used to enhance students’ conceptual understandings at the same time that the students are learning mathematical techniques…
CAS Relation to Student Learning
“ Our research has designed tasks according to the Task-Technique-Theory theoretical framework-- tasks where teachers have contributed to the design, before they have used them in class.
These tasks for Grade 10 algebra engage students in theoretical reflection at the same time that they are learning new paper-and-pencil and CAS techniques.
What we have found is that the process of learning technique in a CAS environment, where theoretical ideas are being co-stimulated, has led to deeper mathematical insights than are the usual for students at this level.”
CAS Relation to Student Learning
Kieran’s research (with Damboise) has illuminated CAS can help “weak” students in both technique and theory. “The CAS provided a certain security to these weak students that led them to examine the CAS outputs for structure and patterns. They were provided with tasks, of course, that encouraged such observations.”
Has Your Concept of CAS Changed?
• 12 commented.
• Some (Mahoney, Hollebrands, Palmiter) indicated no significant change. Fey mentions the basic functionality has remained constant but the interface is more user-friendly.
• Many (Edwards, Goldenberg, McMullin, Waits, Palmiter) cited the move away from “paper-and-pencil” work, the “drudge part,” freeing up time for the “thinking part.”
Has Your Concept of CAS Changed?
• Goldenberg explained: “What I do believe … is that the purpose of such skills as long division and algebraic manipulation has changed, and that the change should be acknowledged in curricula.
• It used to be that we needed these skills for the sake of getting answers. That’s no longer true. If all we want is the answer, use the calculator, especially if the computation is too long or complex without technology.
• Kids no longer need to be super-fast and super-skilled at [make your own list]. But some of these skills are still needed just in order to think about what calculations we’d like the calculator to make.
• If I see no pattern in a sequence of numbers, even roughly, I don’t know what experiment I’d like to perform with the calculator.”
“Value-added” of CAS
Cuoco sees three overlapping use of CAS to help students develop algebraic habits of mind:– Algebraic laboratory: “… making patterns
apparent..”– Algebraic calculator: “… reduces computational
overhead…”– Modeling tool for algebraic structures: “…
medium of expressing abstract algebraic structure.”
Has Your Concept of CAS Changed?
• Drijvers described a gradual change in his conception “from a fascinating expert system in symbolic mathematics into a functionality that is, or should be, part of a whole range of integrated technological tools for teaching and learning mathematics.”
Has Your Concept of CAS Changed?
Jakucyn has “… come to appreciate CAS more for its aid as a pedagogical tool and less as a checker or to simply manipulate the mathematical symbols…”
and finds that “…students who are of average or low mathematics ability can benefit tremendously from using a CAS to develop mathematical concepts and skills.”
Has Your Concept of CAS Changed?
Waits confessed to being “very naïve in the late 1980s” thinking CAS use in schools would follow the growth of graphing technology.
• “Computer graphing was an enhancement of the mathematics and could be used to promote better understanding. ...
• However, computer symbolic algebra was viewed as a replacement of “paper and pencil” manipulative mathematics… which posed a huge threat to teachers” particularly those “who knew nothing about mathematics other than ‘manipulation’” of symbols.
• “At the college level, math prof troglodytes came out of the woodwork against the use of CAS... for many of the same reasons.”
New Developments in CAS?
• 12 responded.
• Several (Drijvers, Mahoney, McMullin, Palmiter, Fey, Hollebrands) mentioned the increase in portability (iPhone apps tablets) and …
• improvements in user-interfaces particularly related to the entry of mathematical expressions.
New Developments in CAS?
• Kutzler expressed skepticism that “what the market would need” will not be met by “the industry’s willingness to respond to this.”
• Steen: “At the college level, faculty have sufficient knowledge and authority to make use of CAS when and if they feel it would be helpful. Even within the same course, the degree of CAS use will differ greatly, but so do many other things. I see this variety as a strength, not as a problem. In the K-12 sector, however, the increasing political pressure for standardization of curricular and testing will create impediments for constructive development of CAS in teaching and learning. CAS needs flexibility to develop effectively in educational contexts, but don’t see how that is compatible with incessant high stakes testing. “
New Developments in CAS?
Drijvers more optimistically expects “CAS to come out of its isolated position, and get more integrated, sometimes visible and sometimes invisible, into other technological environments. …
From the pedagogical perspective, I guess CAS will act rather on the background as an engine behind educational tools for math, than an independent and front-end tool for users. …
The challenge for designers, educators and researchers is to find ways to exploit the potentials of computer algebra.”
New Developments in CAS?
Rich described his work on Rubi: an online, rule-based repository of mathematical knowledge.
“Unlike the current proprietary CAS, such a repository would be an open-source knowledge base accessible to everyone. The knowledge required to automate mathematics will be stored in the form of high-level reduction rules.
These rules are expressed as mathematical formulas with precise application conditions when they are to be applied. Therefore mathematicians can contribute knowledge to the repository in their area of expertise without having to program in a low-level computer language.”
Impediments/Issues Related to CAS
• 15 responses
• While there was overlap two items emerged:
–teachers’ beliefs about learning and –the need for professional
development.
Impediments/Issues Related to CAS
• Related to beliefs, Anonymous mentioned “the most common and strongly held [beliefs] that you have to first master the basics before using technology. This belief constrains imagination about CAS impact.”
• McMullin, Jakucyn, Hollebrands, Palmiter, and Mahoney suggest that new curriculum and state accountability tests might serve to combat the beliefs held by some teachers.
Impediments/Issues Related to CAS
• Goldenberg thinks “… the “game” must be fundamentally changed. Mathematics tends to be about learning things and it needs to be about learning thinking. That’s not news—we’ve all said it, and even hoped that the technology would help force that game change by doing the “drudge work,” thus removing all need to focus on “things” and skills, leaving only the thinking to learn. But it’s not my impression that things have worked out that way. If people still believe that the goal is to get answers, the problems get hairier, and the extra “hair” (arithmetic or algebraic) is relegated to the calculator, but no additional attention is given to the thinking. We’ve simply replaced mental (or paper) skills with button-pressing skills. …
• I think that the major impediment is the public view that mathematics is something most people can’t learn, except perhaps at a skill level, and it is all about application, not reasoning. If the point is to get answers in applications, why not use a tool? And the public is mixed.”
Impediments/Issues Related to CAS
• Edwards, Waits, Jakucyn, Kieran and Saldanha address the importance of professional development.
• Jakucyn states that “… teachers (and parents) are not familiar with the capabilities of a CAS and thus view the use of a CAS as a form of ‘cheating’ …. This may stem from teachers not being trained to use a CAS or from teachers’ and parents’ views of what it means to ‘do the math.’”
Impediments/Issues Related to CAS
Saldanha calls for “more knowledge about what is entailed in incorporating a CAS-oriented perspective” and “… how teachers might restructure instruction in way that support such learning.” He mentions that “few teachers have models of these possibilities, and they need intensive assistance (in the form of resources and training) to develop models and reconceive the mathematics they teach so as to be able to incorporate CAS use in their instruction.”
Impediments/Issues Related to CAS
• Kieran claims that “Most CAS professional development focuses on the button-pushing of the technology and not on the issue of how to construct tasks that will lead to deeper learning of mathematics, with the aid of this technology. Teachers who have tended to use mathematics textbooks as their prime resource for teaching have never really learned how to generate rich mathematical tasks… Teachers need a kind of professional development that will yield a broad vision of the ways in which CAS tasks can be constructed so as to yield the mathematical benefits that the technology can foster.”
Impediments/Issues Related to CAS
Kutzler feels that “computers are an exceptionally wonderful tool for mankind. In the times of DOS, computer use was restricted to techno-phile people. As operating systems improved, computers became more accessible to techno-phobe people.
“Similarly CAS is an exceptionally wonderful tool for math education. But today's CAS products are still only used by techno-phile teachers. We need interfaces on top of CAS engines that make this great technology accessible also to the techno-phobe teachers - and that's by far the biggest crowd.
“I don't see a single tool on the market that makes CAS accessible to a techno-phobe teacher. As long as we need teacher training for CAS products, we are not doing the right thing. How many cell phones would the industry sell if users would need cell phone training before being able to make good use of them?”
Impediments/Issues Related to CAS
• Waits cites the importance of attracting better prepared students to the teaching profession and better professional development on using technology appropriately. He addresses the changing of beliefs by professional development for high school teachers claiming that teachers “feel comfortable and are mostly good at teaching paper and pencil… manipulations (most now obsolete).”
• They feel very threatened if they would have time to teach what we know is really important about mathematical understanding, problem solving, thinking, and reasoning… “things CAS give them time to do.
• “As far as college and universities go: fire-breathing, troglodyte-eating dragons need to be let loose in college mathematics departments!!! And you can quote me!”
Impediments/Issues Related to CAS
• Dick, more calmly, sees promise in the “dynamic linking of objects” that is now available in CAS environments. “By dynamic linking, I mean linkages between and within representations (including symbolic) that are updated instantaneously - changes made to one object or form are immediately reflected in changes to a linked object or form.
• When employed across representations this could allow manipulation on the entry side of numeric data or graphic representations or some virtual physical scenario with the linked result in symbolic form
Impediments/Issues Related to CAS
• Employed within symbolic representations, we'd have two (or more) dynamically linked symbolic forms where a change in one immediately results in a change in the linked form
• To Dick, this opens up 1.new power to ask sense-making cross-
representational questions, and2.new opportunities to actually use CAS as a tool to
gain insight into symbolic processes - transforming it from the black box task servant to a glass box illuminator.
Dynamic Linking Across Representations
Nspire CAS Dynamics
• Media4Math Mini-Tutorial on Quadratic with Sliders
Dynamic Linking Example
By John Hanna
Impediments/Issues Related to CAS
By Peter Flynn
By Thomas Edwards & Asli Özgün-Koca
By John Losse
Resources for Follow-Up
Computer Algebra Systems in Secondary School Mathematics Education
$12(!) from NCTM
Edited by James T. Fey, Al Cuoco, Carolyn Kieran, Lin McMullin, and Rose Mary Zbiek
Resources for Follow-Up
Professional organize to “encourage active interest in the use of CAS..”
June 26-27, 2010 at
New Trier High School, Illinois
Computer Algebra Systems (CAS): From Where Did They Come,
and Where Might They Go?
• Questions or Comments.?
• Powerpoint and References available at www.ite.sc.edu/dickey.html